What are isomorphisms in different categories of vector spaces and algebras?

Question

I start this post to collect different categories of vector spaces and operator algebras and the isomorphisms in them.

There are a variety of vector spaces (normed, Banach, Hilbert etc.) and a variety of operator algebras (Banach, C*, von Neumann etc.). There are also tons of different maps that are used between any two algraic structures (isometry, isomorphism, $*$-homomorphism etc.). So which of these maps are isomorphisms in the categorical sense? The list is open to any category that is used in functional analysis or operator algebras. I am aware that the same class of objects with different morphisms gives us different categories. So if you are talking about some non-obvious morphism, please mention it.

I will add every answer to the following list:

(All maps are linear unless mentioned otherwise.)

1. In the category $Vect_\mathbb{K}$ of vector spaces over a scalar field $\mathbb{K}$, the isomorphisms are simply invertible maps.
2. In the category of normed spaces with bounded maps, isomorphisms are bounded maps with bounded inverses.
3. In the category of normed spaces with contractions, isomorphisms are linear surjective isometries.
4. In the category of Banach spaces with bounded maps, isomorphisms are bounded bijections.
5. In the category of unital $C^*$-algebras with u.c.p. maps, isomorphisms are $*$-isomorphisms.
6. In the category $Rep(A)$ of representations of a C*-algebra $A$, isomorphisms are unitary equivalences. (Note that two representations are unitarily equivalent if there exists a bijective bounded intertwiner which need not be unitary.)
7. In the category of von Neumann algebras with normal $*$-homomorphisms, isomorphisms are $*$-isomorphisms.

and so on...

Answer 1

In the category of normed space with continuous linear maps isomorphisms are continuous linear maps with continuous inverse.

In the category of normed spaces with linear contractions ($\|T(x)\|\le\|x\|$) isomorphisms are surjective linear isometries.

In the category of Banach spaces and continuous linear maps isomorphisms are continuous linear bijections (because of the closed graph theory).

Answer 2

• In the category of $W^*$-algebras with normal $*$-homomorphisms as morphisms, isomorphisms are $*$-isomorphisms (a $*$-isomorphism between two $W^*$-algebras is automatically normal!).

• In the category of unital $C^*$-algebras with unital completely positive maps as morphisms, the isomorphisms are $*$-isomorphisms. It may not be obvious that an isomorphism in this category preserves the multiplication, but this follows by a smart application of Kadison's inequality for ccp maps.