%\chapter{Errata for Volume I} \documentstyle[12pt]{article} \input amssym.def \input amssym %%%%%%%real, complex, integers, natural numbers, one-torus, field, disc \newcommand{\real}{\Bbb R} \newcommand{\complex}{\Bbb C} \newcommand{\Zahl}{\Bbb Z} \newcommand{\whole}{\Bbb N} \newcommand{\torus}{\Bbb T} \newcommand{\field}{\Bbb F} \newcommand{\disc}{\Bbb D} \def\trinorm#1{\setbox0 = \hbox{$\vert\vert\vert$} \copy0 #1 \box0} \newcommand{\wig}[1]{{\cal #1}} \newcommand{\atimes}[1]{\vbox{\hbox{\kern1pt\lower5pt\hbox{$#1$}}\hbox{$\otimes$}}} \newcommand{\mapright}[1]{\smash{\mathop{\longrightarrow}\limits^{\scriptstyle{#1}}}} \newcommand\proof{ \par \noindent {\bf \em Proof}\ \ } \def\norm#1{\Vert#1\Vert} \textwidth = 6.5 in \textheight = 9.0 in \oddsidemargin = .0 in \evensidemargin = .0 in \topmargin = 0.0 in \headheight = 0.0 in \headsep = 0.0 in \baselineskip 20pt \begin{document} \pagestyle{plain} \begin{center} {\bf Banach Algebras and the General Theory of *-Algebras\\ by Theodore W. Palmer\\ Errata for Volume I: Algebras and Banach Algebras, June 10, 1997} \\ \end{center} \vskip15pt \noindent P.\ 11 line 8: Replace ``2.2.1'' by ``$2.2.2$''. \vskip5pt \noindent P.\ 16 line 27: Replace ``$\sigma$'' by ``$\sigma^\infty$''. \vskip5pt \noindent P.\ 19 lines -3, -2: Interchange ``right '' and ``left''. \vskip5pt \noindent P.\ 20 lines 7, 8: Replace ``on $\wig A$, any regular norm $\trinorm \cdot$ on $\wig A^1$ which extends $\norm\cdot$ on $\wig A$ satisfies:'' by ``on $\wig A$. Any regular norm $\trinorm \cdot$ on $\wig A^1$ which extends $\norm\cdot$ on $\wig A$ satisfies the first two inequalities below. If $(\wig A,||\cdot||)$ is complete, the third inequality also holds.''. (Noted by Haresh V.\ Dedania who showed by example that the third inequality can fail in the incomplete case. His example involves the non-unital subalgebra $\wig A=\{f\in A({\Bbb D}): f(1)=0\}$ of the disc algebra which has a unital completion in the incomplete regular norm $||f||=\sup\{|f(\lambda)|: |\lambda|\le 1/2\}$. For the other regular norm on $\wig A^1\simeq A({\Bbb D})$ use $|||\lambda +f|||= \sup\{|\lambda + f(\mu)|:|\mu|\le 1/2\}$.)\vskip5pt \noindent P.\ 30 line -2: Replace ``1.7.1'' by ``1.7.14''. \vskip5pt \noindent P.\ 32 line 24: After the comma add ``where we must suppose that $\rho$ is an isometry and $\tau$ is contractive,''. \vskip5pt \noindent P.\ 35 line -5: Replace ``regular homomorphism and'' by ``regular homomorphism, $\rho({\cal A})$ is closed in ${\cal D}({\cal A})$ and'' (Noted by Olaf Berndt.) \vskip5pt \noindent P.\ 38 lines 7, 8: My remark here is totally false. There is no {\em algebra} homomorphism which makes the coproduct diagram commutative. What I claimed is true for linear spaces or modules, but not for algebras. There are coproducts in both the topological and geometric categories of commutative Banach algebras. We describe both at once, leaving the details to the reader. (Olaf Berndt showed me this construction in 1995.) Let ${\cal A}^\#$ and ${\cal B}^\#$ denote the commutative Banach algebras $\cal A$ and $\cal B$ with units adjoined even if they are already unital. Use the norm $||(\lambda, a)||= |\lambda| +||a||$. The coproduct ${\cal A}\amalg{\cal B}$ of $\cal A$ and $\cal B$ is the kernel of the map $\chi\colon {\cal A}^\#\hat{\otimes}{\cal B}^\#\rightarrow{\Bbb C}$ from the projective tensor product, defined by $\chi((\lambda, a)\otimes(\mu, b))= \lambda\mu$. The injection maps are $\eta_1(a)= a\otimes 1$ and $\eta_2(b)=1\otimes b$. Because of commutativity, these satisfy the universal property of a coproduct. It appears to be unknown whether there is a coproduct in any useful category of not-necessarily-commutative algebras. (Error noted by Paul M.\ Cohn and Olaf Berndt.) \vskip5pt \noindent P.\ 42 line 4: Replace ``$\varphi_{\alpha}$'' by ``$\varphi_{\beta\alpha}$''. \vskip5pt \noindent P.\ 44 line 8: Replace ``$\varphi_{\alpha}$'' by ``$\varphi_{\beta\alpha}$''. \vskip5pt \noindent P.\ 44 line 12: Replace ``$\beta\ge a$'' by ``$\beta\ge \alpha$''. \vskip5pt \noindent P.\ 139 line -6: Replace ``this topology.'' by ``the weak* topology.''. \vskip5pt %( \noindent P.\ 143 line 20: Remove extra ``)''. \vskip5pt \noindent P.\ 144 line -5: The first ``$du$'' should be ``$dv$''. \vskip5pt \noindent P.\ 149 line -3: Interchange the subscripts ``$G$'' and ``$H$''. \vskip5pt \noindent P.\ 154 line -5: Replace ``subalgebra of $L^1(G)$'' by ``algebra''. \vskip5pt \noindent P.\ 164 line 7: Add ``$k\not= \ell$''. \vskip5pt \noindent P.\ 164 line 13: Replace ``{\em (16)}'' by ``(16)''. \vskip5pt \noindent P.\ 171 line -11: Replace ``{\em Theorem 1.10.6}'' by ``Theorem 1.10.6''. \vskip5pt \noindent P.\ 173 line 18: Replace ``$g(k^{-1}h)$'' by ``$\hat g(k^{-1}h)$''. \vskip5pt \noindent P.\ 177 lines 4, 5: Replace ``a Banach space $\wig X$ satisfying $\wig X\hat\otimes\wig X=\wig X\check\otimes\wig X$'' by ``an infinite dimensional Banach space $\wig X$ with $\wig X\hat\otimes\wig X$ and $\wig X\check\otimes\wig X$ naturally isomorphic''. \vskip5pt \noindent P.\ 179 line -6: Change ``be'' to ``by''. \vskip5pt \noindent P.\ 184 lines 16, 18 and 19: The second ``${\cal A}^{(1)}$'' should be ``${\cal A}^{(2)}$'' each time. Also ``the'' should be ``then'' on line 19. \vskip5pt \noindent P.\ 222 line -3: Replace ``Kaplansky'' by ``Kakutani''. \vskip5pt \noindent P.\ 223 line 12: Replace ``$T_n-T$'' by ``$T-T_n$'' twice. \vskip5pt \noindent P.\ 223 line 15: Remove ``by''. \vskip5pt \noindent P.\ 233 line 20: Immediately after Definition 2.4.1 it would be good to include the following corollary of the Gelfand--Mazur Theorem (2.2.3): {\bf Corollary}\ \ {\em A division algebra is isomorphic to $\complex$ if and only if it is a spectral algebra.} \vskip5pt \noindent P.\ 237 line 15: Insert ``maximal'' before ``ideals''. \vskip5pt \noindent P.\ 238 line 7: Replace ``[1977a] [1977b]'' by ``[1977]''. \vskip5pt \noindent P.\ 238 line 13: Remove ``M. I.''. \vskip5pt \noindent P.\ 239 line -5,-4: Replace ``$(\lambda_0 - a)^{-1}$ has the decomposition $\lambda_0^{-1} + c$ with $c\in\wig A^1$.'' by ``$(\lambda_0 - a)^{-1}\in\wig A^1$ has the decomposition $\lambda_0^{-1} + c$ with $c\in\wig A$.''. \vskip5pt \noindent P.\ 240 line -18 to -3: Replace by $$\mbox{``}\sigma(a) = C^2\sup\{\rho(a + b) - \rho(b): b\in\wig A\}\qquad \forall\ a\in\wig A.$$ Then $\sigma$ satisfies $C^2\rho(a) \le \sigma(a)\le C^3\rho(a)$ for all $a\in\wig A$ and \begin{eqnarray*} \sigma(a + c) & = & C^2\sup\{(\rho(a + c + b) - \rho(c + b)) + (\rho(c + b) - \rho(b)): b\in\wig A\}\\ & \le & \sigma(a) + \sigma(c)\qquad \qquad \qquad \qquad \qquad \quad\ \ \forall\ a,\,c\in\wig A;\\ \sigma(ab) & \le & C^3\rho(ab)\le C^4\rho(a)\rho(b)\le \sigma(a)\sigma(b)\qquad \forall\ a,\,b\in\wig A. \end{eqnarray*} Hence $\sigma$ is a spectral semi-norm. Also, the ideal on which $\sigma$ vanishes is the largest ideal on which the spectral radius vanishes. Corollary 2.3.4 shows that it is the Jacobson radical. Thus $\sigma$ induces a norm $\norm\cdot$ on $\wig A/\wig A_J$ defined by $\norm{a + \wig A_J} = \sigma(a)$ for all $a\in\wig A$. However, this implies \begin{eqnarray*} \norm{(a + \wig A_J)(b + \wig A_J)} & = & \sigma(ab)\le C^3\rho(ab) = C^3\rho(ba)\le C\sigma(ba)\\ & = & C\norm{ba + \wig A_J} = C\norm{(b + \wig A_J)(a + \wig A_J)}. \end{eqnarray*} If $\wig A$ is unital, the last lemma shows that $\wig A/\wig A_J$ is commutative. When $\wig A$ is not unital we also need Proposition 2.2.1(e) to show \begin{eqnarray*} \sigma^1((\lambda + a)(\mu + b)) & = & |\lambda\mu| +\sigma(\lambda b+ \mu a + ab)\qquad\quad\forall\ \lambda+a, \mu +b\in\wig A^1\\ & \hskip-30pt \le & \hskip-17pt C^3(|\lambda\mu| +\rho(\lambda b+ \mu a + ab)) \le3C^3\rho((\lambda + a)(\mu + b))\\ & \hskip-30pt = & \hskip-17pt 3C^3\rho((\mu + b)(\lambda + a)) \le 3C\sigma^1((\mu + b)(\lambda + a)).\mbox{''.} \end{eqnarray*} \vskip5pt \noindent P.\ 241 line 4: Replace ``$p$'' by ``$\sigma$''. \vskip5pt \noindent P.\ 241 line -3: Replace ``algebra'' by ``spectral algebra''. \vskip5pt \noindent P.\ 253 line -10: ``and hence'' should be ``{\em and hence}''. \vskip5pt \noindent P.\ 289 line 2: Change ``$\wig S(\wig X)$'' to ``$\wig B(\wig X)$''. \noindent P.\ 292 line 4: Change ``$(\lambda_0-K)^*\wig X$'' to ``$(\lambda_0-K^*)^a\wig X$''. \noindent P.\ 294 line 18: Change ``Alan'' to ``Allen''. Also the unpublished notes I quote were published as Pearcy and Shields [1974]. \vskip5pt \noindent P.\ 294 line 21: Change ``$K$'' to ``$\wig X$''. \vskip5pt \noindent P.\ 295 line 13: Change ``$\sum_{j=1}^n$'' to ``$\prod_{j=1}^n$''. \vskip5pt \noindent P.\ 297 line 14: Change ``Jorg\'e'' to ``J\"org''. \vskip5pt \noindent P.\ 300 line -7: Remove ``.'' (period). \vskip5pt \noindent P.\ 307 line -4: Replace ``Proposition 3.1.2(b)'' by ``Theorem 3.1.2(b)''. \vskip5pt \noindent P.\ 308 line 5: Replace ``algebra'' by ``spectral algebra''. \vskip5pt \noindent P.\ 309 lines -15, -14: Replace ``Either the Gelfand space is empty (which occurs if and only if $\wig A$ is a Jacobson-radical algebra)'' by ``Either $\wig A$ is a Jacobson-radical algebra (so its Gelfand space is empty)''. (The beginning of the proof shows that an almost commutative spectral algebra has empty Gelfand space if and only if it is Jacobson-radical.) \vskip5pt \noindent P.\ 310 line 4: Replace ``$|\hat{a_n}|$'' by ``$|\gamma_n(a)|$''. \vskip5pt \noindent P.\ 311 lines -14: Replace ``4.2.15'' by ``4.2.18''. \vskip5pt \noindent P.\ 311 lines -5: Remove ``, We give related results in Theorem 4.2.15''. \vskip5pt \noindent P.\ 314 line 1: Replace ``2.1.5'' by ``2.5.15''. \vskip5pt \noindent P.\ 321 line 12: Replace ``the last statement of the theorem holds.'' by ``$\omega_{h_\alpha}$ converges in the Gelfand topology.''. \vskip5pt \noindent P.\ 321 line 18: Replace ``$\Gamma_{\cal A^1}^0$'' by ``$\Gamma_{\cal A^1}$''. \vskip5pt \noindent P.\ 322 line -13: Replace ``$\Gamma_{\cal B}$'' by ``$\Gamma_{\cal A}$''. \vskip5pt \noindent P.\ 324 line -7: Replace ``$\gamma_2(a)$'' by ``$\gamma_2(\varphi(a))$''. \vskip5pt \noindent P.\ 324 line -3: After ``Define'' insert ``$\tilde \gamma\colon \varphi(\wig A)\rightarrow \complex$ by $\tilde \gamma=\gamma\circ\varphi^{-1}$ and''. Then change the next seven undecorated occurrences of $\gamma$ to $\tilde\gamma$. \vskip5pt \noindent P.\ 325 line 1: Replace ``$\overline{\gamma}(a)$'' by ``$\overline{\gamma}(\varphi(a))$''. \vskip5pt \noindent P.\ 326 line 4: Replace ``.'' (period) by ``,'' (comma). \vskip5pt \noindent P.\ 333 line 2: Replace ``$e^{nt}$'' by ``$e^{int}$''. \vskip5pt \noindent P.\ 339 line 18: Replace ``$\gamma\rightarrow K$'' by ``$\gamma\colon G\rightarrow K$''. \vskip5pt \noindent P.\ 339 line 22: Replace ``$U_G$'' by ``$\overline{U_G}$''. \vskip5pt \noindent P.\ 340 line -7: Replace ``$(ac\otimes d)$'' and ``$(c\otimes bd)$'' by ``$\theta(ac\otimes d)$'' and ``$\theta(c\otimes bd)$'', respectively. \vskip5pt \noindent P.\ 345 lines -2, -1: Replace ``Hence, if $\Gamma_{\wig A^1}$ is nonempty, the Gelfand transform satisfies $$\widehat{I_a^\Gamma(f)} = f\circ \hat{a}.\hbox{''}$$ by ``Hence, the Gelfand transform on $\wig A^1$ satisfies $$\widehat{I_a^\Gamma(f)} = f\circ \hat{a}\qquad\forall\ a\in\wig A.\hbox{''.}$$ \vskip5pt \noindent P.\ 346 line -2: Remove close parenthesis just before ``$d\lambda$''. \vskip5pt \noindent P.\ 361 line 15: Replace ``$||a||^n$'' by ``$||a^n||$''. \vskip5pt \noindent P.\ 361 line -8: Replace ``$1-\rho(b)\le \sqrt{1-\rho(a)}<1$'' by ``$\rho(b)\le 1-\sqrt{1-\rho(a)}$''. \vskip5pt \noindent P.\ 363 line 8: Replace ``$\wig F_b$'' by ``$\wig F_a$''. \vskip5pt \noindent P.\ 365 lines 14, 15: Replace ``$J_a(\wig F_a)$'' and ``$I_a(\wig F_a)$'' by ``$J_a(e)$'' and ``$I_a(e)$'', respectively. \vskip5pt \noindent P.\ 372 line -19: The inequality should read ``$\omega_m\omega_n\ge\omega_{m+n}$''. \vskip5pt \noindent P.\ 408 lines -11, -12: Delete the last sentence of Corollary 3.5.14 (a). (Noted by Michael A. Dritschel.)\vskip5pt \noindent P.\ 408 line -2: Delete the words ``satisfying $e{\cal A}\cap{\cal A}_J=\{0\}$''.\vskip5pt \noindent P.\ 421 line 12: Replace ``the last statement of the theorem holds.'' by ``it converges as described in the penultimate sentence.''.\vskip5pt \noindent P.\ 422 line 1: Replace ``$h(u)$'' by ``$h(v^{-1})$''.\vskip5pt \noindent P.\ 426 line 2: Replace ``is'' by ``as''.\vskip5pt \noindent P.\ 426 lines 10, 12: Replace ``$\eta_p$'' by ``$\tau_p$'' three times.\vskip5pt \noindent P.\ 428 line 4: Replace ``$B(\hat G)$'' by ``$A(\hat G)$''.\vskip5pt \noindent P.\ 428 line 14: Replace ``3.6.9'' by ``3.6.8''.\vskip5pt \noindent P.\ 428 line -15: Replace ``(1)'' by ``(24)''.\vskip5pt \noindent P.\ 428 line -11: Replace ``(2)'' by ``(25)''.\vskip5pt \noindent P.\ 429 line 5: Replace ``(3) and (4)'' by ``(26) and (27)''.\vskip5pt \noindent P.\ 429 line 10: Replace ``(6)'' by ``(29)''.\vskip5pt \noindent P.\ 431 line 10: Replace ``(6)'' by ``(29)''.\vskip5pt \noindent P.\ 431 line 14: Replace ``(18)'' by ``(30)''.\vskip5pt \noindent P.\ 432 line 13: Replace ``Stiltjes'' by ``Stieltjes''.\vskip5pt \noindent P.\ 433 line 3: Replace ``$L^1(G)$'' by ``$L^1(\hat G)$''.\vskip5pt \noindent P.\ 435 line -22: Replace ``idemopotent'' by ``idempotent''.\vskip5pt \noindent P.\ 435 line -16: Replace ``another'' by ``Another''.\vskip5pt \noindent P.\ 440 line -20: Bad margin. Remove the word ``will'' from the previous line to correct this. \noindent P.\ 448 line 13: Replace the last ``${\cal P}_2{\cal I}$'' by ``${\cal P}_1{\cal I}$''.\vskip5pt \noindent P.\ 457 line -5: Replace ``${\cal X}$'' by ``${\cal X}_1$''.\vskip5pt \noindent P.\ 459 line 3: Replace ``Theorem 4.1.3 and Proposition 4.1.5'' by ``Theorems 4.1.3 and 4.1.6''.\vskip5pt \noindent P.\ 63 line -3: I wish I had added another one sentence paragraph to Theorem 4.2.11 ``Hence a primitive spectral algebra is unital if it has a nonzero center.''. At the end of the proof I could say: ``The last sentence is obvious.''.\vskip5pt \noindent P.\ 477 line -1: Replace ``$a\in{\cal A}$'' by ``$a\in{\cal I}$''.\vskip5pt \noindent P.\ 479 line -12: Replace ``Proposition 2.5.3(e) gives'' by ``Proposition 2.5.3(c) and Theorem 4.3.6(e) give''.\vskip5pt \noindent P.\ 498 line -11: Replace ``these notes'' by ``this work''.\vskip5pt \noindent P.\ 504 line -19: Replace ``[1898]'' by ``[1894]''.\vskip5pt \noindent P.\ 512 line -7: At the beginning of the proof add: ``Note that (d) excludes one- or two-sided identities.''. \vskip5pt \noindent P.\ 513 line 12: Replace the first sentence of the paragraph by: ``Let ${\cal S}$ be the free semigroup $FS(2)$ described in the last example with the identity element removed.''. Then replace each $FS(2)$ by $\wig S$. (Noted by Haresh V.\ Dedania.) \vskip5pt \noindent P.\ 513 line -11: Replace ``$f(n)V^nf(m)$'' by ``$f(n)V^n(g(m))$''.\vskip5pt \noindent P.\ 518 line -9: Replace ``1.1.16'' by ``1.2.10''. \vskip5pt \noindent P.\ 519 line -1: There should be a space after ``S.''. \vskip5pt \noindent P.\ 528 lines -11, -3: Replace ``$\int_U\lambda(U)^{-1}$'' by ``$\lambda(U)^{-1}\int_U$'' both places.\vskip5pt \noindent P.\ 530: Replace lines 11--13 by: ``note that $G$ is unimodular and $e_U$ satisfies $e_U(uv)=e_U(u(vu)u^{-1})=e_U(vu)$. Thus arbitrary $f\in L^1(G)$ satisfy $$ e_U*f(u) = \int e_U(uv)f(v^{-1})dv\\ = \int \Delta(v^{-1})f(v^{-1})e_U(vu)dv = f*e_U(u).\hbox{ ''.}$$ \noindent P.\ 532 lines -2, -1: Replace ``$\langle \ \norm{ue - e} < \varepsilon$ / $ \norm{eu - e} < \varepsilon\ \rangle$'' by \linebreak ``$\langle \ \norm{ua - a} < \varepsilon$ / $ \norm{au - a} < \varepsilon\ \rangle$''.\vskip5pt \noindent P.\ 534 line 16: Replace ``$\overline{\rm span}(T_{\cal A}{\cal X}^j)$'' by ``$\overline{\rm span}(T^{(j)}_{\cal A}{\cal X}^j)$''.\vskip5pt \noindent P.\ 543 line 4: Replace ``${\cal M}$'' by ``${\cal L}$''.\vskip5pt \noindent P.\ 549 line 18: Replace ``[1993]'' by ``[1997]''.\vskip5pt \noindent P.\ 550 line -4 of text: Replace ``$\overline{\rm span}$'' by ``span''.\vskip5pt \noindent P.\ 553 line 9 of text: Replace ``derivation'' by ``homomorphism''.\vskip5pt \noindent P.\ 560 line 6: Change ``3.4.18'' to ``3.4.22''.\vskip5pt \noindent P.\ 560 lines -19, -18: Delete``(if one assumes the continuum hypothesis),''. (This and the next correction noted by Volker Runde.)\vskip5pt \noindent P.\ 561 line -18: Replace ``Theorem 2.5.15'' by ``Proposition 2.5.16''.\vskip5pt \noindent P.\ 570 line -2: Replace ``no'' by ``every (or some)''. \vskip5pt \noindent P.\ 574 line 7: Replace ``elements'' by ``hermitian elements''. \vskip5pt \noindent P.\ 576 line 19: Replace ``in any commutative Banach algebra.'' by ``in any Banach algebra. (They said, but did not use, ``commutative'' in their hypotheses.)''. \vskip5pt \noindent P.\ 576 line -12: Replace ``\S3.4.16'' by ``\S3.4.20''. \vskip5pt \noindent P.\ 576 line -3: Replace ``Our results in Example 3.2.15 show that'' by ``As noted in Example 3.2.15,'' and delete next ``that''. \vskip5pt \noindent Pp.\ 578 and 579: Replace all occurrences of ``ZFC'' by ``ZF''. \vskip5pt \noindent P.\ 594 lines 20, 21: Interchange ``derivations'' and ``homomorphisms''. \vskip5pt \noindent P.\ 603 line 17: Replace ``$(\delta(\wig P) + \wig P)\wig P = \{0\}$'' by ``$(\delta(\wig P) + \wig P)/\wig P = \{0\}$''. \vskip5pt \noindent P.\ 605 line 17: Replace ``$\rightarrow$'' by ``$\mapsto$''. \vskip5pt \noindent P.\ 621 line 1: The first ``$\tilde{h}k$'' should be ``$k\tilde{h}$''. \vskip5pt \noindent P.\ 669 line 13: Replace ``projections'' by ``idempotents''. \vskip5pt \noindent P.\ 704 line 13: Replace ``invariant'' by ``irreducible''. \vskip5pt \noindent P.\ 716 lines 1, 3: Change Jorg\'e to J\"org twice. \vskip5pt \noindent P.\ 717 line 25: ``Ardzewski'' should be ``Nardzewski''. \vskip5pt \noindent P.\ 787: After the entry ``norm'' there should be a parenthetical remark ``(see also tensor norm)''. \vskip5pt \begin{center} {\bf If you find additional errors, please send them either by E-mail}\\ (palmer@math.uoregon.edu) or to Theodore W.\ Palmer, Department of \\ Mathematics, University of Oregon, Eugene, Oregon 97403, U.\ S.\ A. \end{center} \end{document} \end