By Kehe Zhu

An advent to Operator Algebras is a concise text/reference that specializes in the elemental ends up in operator algebras. effects mentioned contain Gelfand's illustration of commutative C*-algebras, the GNS development, the spectral theorem, polar decomposition, von Neumann's double commutant theorem, Kaplansky's density theorem, the (continuous, Borel, and L8) useful calculus for regular operators, and kind decomposition for von Neumann algebras. workouts are supplied after every one bankruptcy.

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**Extra resources for An Introduction to Operator Algebras**

**Example text**

There must exist a - cp(x)l. Clearly cp(xo) vector unit == O. 1 we imply that Xo =

1, there exists no multiplicative linear functional on the of the linear functional trace, by uniqueness any multiplicative M n (C) must be a constant multiple of the trace.

In N if x yx)2, x(yxy) + (yxy)x) is. Since) (xy- 'IIx)2 + (XU + lIx)2 = 2 [x(yxy) + (yxy)x]))) that the 26) Linear Multiplicative Functionals) and) - (cp(xy yx))2 = cp((xy is in N for all x we see that xy - yx and xy + yx, we concludethat xy belongs particular, N is closedunder multiplication, yx)2), and yEA. Adding to N for all x in N and y the proof of the completing E N xy - in A. yx In theorem. 1 Show M n (C) has no nontrivial ideals. that maximal ideals. 3 Convince yourself linear functionals on in D.

6 SupposeA is a Banach alRebra and x, YEA. 1 Let T be the = on H defined operator Tf(x) \302\2432 [0, xE = xf(x), 1] by [0,1]. is the spectrum of T in B (H)? Generalize your result. Hilbert space. Show that H is a (separable)infinite dimensional Suppose for every nonempty compact set K in the complex plane there exists an = K. 5 If x the spectrum that a(T*) is invertible of T = in {X in defined a Banach 1 x be an {O, ai, a2, . }. B ( Z2). : A E a(x- by a2, . })= ) for every T algebra A, then -1 : A E = a(T)} {A in B(H).